Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,20)(14,15)(16,19)(17,21)(18,23)(22,24)(25,26)(27,32)(28,37)(29,30)(31,34)(33,40)(35,39)(36,38);(1,2,6,8)(3,5,7,4)(10,15)(12,21)(13,16)(18,24)(25,27,31,28)(26,32,34,37)(29,38,33,35)(30,36,40,39);(1,3,6,7)(2,4,8,5)(9,11,14,19)(10,13,15,16)(12,18,21,24)(17,22,20,23)(25,28,31,27)(26,32,34,37)(29,38,33,35)(30,39,40,36);(1,4,6,5)(2,3,8,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,22)(21,24);(1,2)(3,4)(5,7)(6,8)(9,12,14,21)(10,17,15,20)(11,18,19,24)(13,22,16,23)(25,29,31,33)(26,30,34,40)(27,35,28,38)(32,39,37,36);(1,5,6,4)(2,7,8,3)(9,13,14,16)(10,11,15,19)(12,22,21,23)(17,18,20,24)(25,30)(26,33)(27,36)(28,39)(29,34)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662317);